1. Introduction

Laser-induced filamentation damage in optical components bulk is a key limiter of high-power laser systems, motivating the study of potential mitigation methods (e.g., for Mega-Joule class laser systems) [1,2]. Filamentary damage in an optic’s bulk is a result of self-focusing, a process in which the Kerr nonlinearity exceeds diffraction and light self-traps into a rapidly tightening spot until the intensity is high enough to generate damage. The damage formation process, once the required intensity has been exceeded, involves multi-photon ionization and generation of a carrier density in the bulk which in turn limits the intensity growth. The physics of damage formation is discussed elsewhere [3–10] and is beyond the scope of this work. Here, we focus on extending the distance to collapse before self-focusing is reached, and thus the underlying physics can be described by the nonlinear Schrödinger wave equation (NLSE).

The idea of splitting an optical element into segments spaced by airgaps in order to suppress self-focusing was proposed and studied [10–13]. The proposed mechanism is allowing the beam (or alternatively an isolated hotspot on a beam) that is self-focusing at the glass segments to diffract and expand in the airgap between the two thinner optics, providing a system for diffraction management which can help avoid collapse. Therefore, in a setting of a single growing perturbation on a beam, increasing the airgap size serves as a monotonically and limitless method to mitigate self-focusing, as the beam is further expanded in the airgap between the optics segments. However, for high-power lasers a more typical seed is closely packed perturbations on a large aperture beam (typically apodised top-hat or super-Gaussian) where the hot-spots on the beam grow into filaments [14], and thus it is unclear what is the role of the neighboring hot spots expansion and interference. The effect of this beam contrast interference on the airgap mitigation method is the focus of this study.

This paper is organized as following: in section 2 we study the effect extending the airgap has on total permissible propagation distance in glass before collapse, using a periodic intensity perturbation excitation. While this excitation source term is less realistic than a one having a random perturbation, its behavior is more easily trackable and thus provides an instructive tool for understanding the interaction between the interfering hot spots (as a note, finishing errors on large optics sometimes leads to repetitive patterns, and therefore this discussion might have some practical implications as well). Thereafter, in section 3, filamentation in an airgap system is studied for a flat-top beam with a random perturbation noise source. The underlying physics is probed by a parametric study followed by a conclusion section 4. Even though we refer in this study to a material system and dimensions especially relevant to Mega-Joule class high power lasers, the conclusions at large apply to other cases of high-power lasers (e.g., high power Peta-Watt class lasers, high power lasers for directed energy applications) and to more general beam contrast collapse and evolution in alternating Kerr nonlinear media and free-space gaps.

2. Periodic harmonic perturbation excitation

2.1 Method of the numerical calculation

The numerical calculation is based on a beam propagation method (BPM) implementation using a split-step algorithm [15–18] of the nonlinear wave equation, following the same methodology detailed for the same case without airgap in [19]. The governing nonlinear Schrödinger wave equation (NLSE) for the phasor representation of the E-field (harmonic in time, $E \propto \exp \, (i\omega \cdot t - ik \cdot z)$), assuming paraxial propagation along z direction:

This work focuses on the physics behind spatial self-focusing initiated by statistical perturbation on the beam, and utilizing air gap as a diffraction management method for enhancing the distance to collapse. The onset of the damage mechanism is dominated by peak intensity of the pulse, which justifies the spatial model used here. For ultra-short fs pulse systems or for longer pulse system with rapidly modulated pulse shape, temporal mechanisms should be included, e.g.: group velocity dispersion, temporal Kerr self-focusing, and dynamics of free carriers coupled with the beam (e.g., [15]).

The excitation is of a 2D harmonic amplitude perturbation on a spatially invariant intensity, representing the top-hat region of a base beam. For each simulation the parameters that are being changed are: the base beam intensity I₀, the perturbation amplitude a, and the perturbation period W (same period is assumed in both x and y axis), as illustrated in Fig. 1(a). The expression for the input field is:

 

Fig. 1. Numerical calculation of self-focusing length in glass (L_fil) for 2D harmonic perturbation on a flat base beam as a function of the airgap size: (a) illustration of the excitation normalized intensity distribution (I(x,y)/I₀) and the three parameters that are being varied: I₀, a, and W. (b) Illustration of the airgap split glass configuration studied with BPM. (c) Simulated self-focusing length as a function of airgap size (I₀ = 8 GW/cm², a = 0.25, W = 0.2 mm, L_in=2 cm).

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We study the effect of the airgap size, L_gap, on the total propagation distance in glass till self-focus, L_fil, as illustrated in Fig. 1(b). A constant set of parameters is chosen for the excitation in this case study {I₀ = 8 GW/cm², a = 0.25, W = 0.2 mm}, and the first glass width is set to L_in=2cm. The airgap is varied for the set of 40 simulations between 0 and 10 cm. For each simulation, the varying medium is implemented in the BPM as the refractive index (linear and nonlinear) used accordingly during the propagation along the z-axis, where n = 1 and n₂ = 0 are assumed for the airgap (i.e., vacuum values, for simplicity), and Fresnel reflections are neglected. A prototypical maximal intensity is illustrated in Fig. 1(b), increasing in the glass segments due to self-focusing. The propagation calculation at the 2^nd glass stops when the peak intensity reaches an arbitrary high value, determining L_fil for this case (i.e., being calculated as the total calculation distance till collapse, L_total, minus L_gap). We choose here an arbitrary value of the filamentation threshold intensity I_fil = 150 GW/cm², which is more than an order of magnitude higher than the initial base beam intensities examined here. Since the self-focusing in the NLSE is not limited (unlike the practical case, where carrier density forms at high intensity which limit the intensity growth through deflection and absorption), the growth is exponential, and when it reaches intensities substantially higher than the initial one, the growth is very rapid, and the choice of L_fil has small sensitivity to the particular arbitrary threshold selection. This methodology is a common practice in this field [10,22].

We have chosen the parameter space for this study to explore regions where mega-Joule systems present more likelihood to filament {I₀ = 6 -10 GW/cm², a = 0.1 - 0.3, W = 0.15 - 0.3 mm, L ∼ few cm} [1,2,19]. Nevertheless, the underlying mechanisms detailed here should apply to other cases of high-power lasers beyond the parameter space of this study.

2.2 Results and discussion

The behavior for a periodic perturbation excitation is fundamentally different from the one for the more commonly discussed for a single hotspot on a beam (or a whole beam) collapse, as the L_fil has limited growth and periodic dependence on the airgap size and not monotonically growing with L_gap. A single hot spot is expected to self-focus in the first glass, and then continue to focus in the airgap (similarly to a case where a beam obtains a focus in the air after a lens) and thereafter expand in the airgap, and self-focus again at the 2^nd glass. Therefore the behavior for the cases of short airgaps (below 3cm) is following this single hot-spot prediction: for very small airgap (0.25cm) the beam is further focusing in air, and thus the L_fil slightly reduces, but after passing this focal point the hot-spot expands in the air increasingly with the airgap size and so is the L_fil. The dependence of the propagation distance to collapse in the second glass on the perturbation amplitude, width and phase is discussed extensively in [19]. The rapid growth and decay in a periodic manner, observed in Fig. 1(c) at larger L_gap values, differ significantly from this scheme, and questions the feasibility of airgap as a practical mitigation method.

To better understand this periodic behavior, the intensity distribution evolution was tracked with the beam propagation, and representative snapshots are illustrated in Fig. 2 (see schematics of the problem in the inset). Since for all the cases studied in this example (both in Fig. 1(c), and in Fig. 2) the 1^st glass length is the same, L_in = 2cm, the intensity distribution evolution is common to all cases, and includes self-focusing, as illustrated in Fig. 2(a). For five cases of different airgap size of L_gap= 0 (no airgap), 2cm, 3cm, 6cm, and 9cm, the evolution beyond z = 2cm is illustrated in Figs. 2(b)–2(f), accordingly (where the case depicted in Fig. 2(f) is further illustrated by Visualization 1). For the reference case of no airgap (Fig. 2(b)) the self-focusing is continuing undisruptive, leading to exponential growth of the maximal intensity, reaching L_fil=3.48cm. When the airgap is relatively small (L_gap=2cm, Fig. 2(c)), the dynamics follows the simplified picture of airgap mitigation, as the hot-spots diffract and expand with the propagation in the airgap, and at the re-entrance to the 2^nd glass the self-focusing restart, resulting in a delayed self-focusing in glass, and L_fil=4.11cm – larger than the no airgap reference case. This modest change in L_fil is expected to follow the diffraction behavior, thus, to expand more rapidly past the Rayleigh distance, and become larger for narrower perturbations.

 

Fig. 2. Snapshots along propagation of normalized intensity distributions (I/I₀) for different airgap size (periodic excitation – one spatial period is simulated and shown). The schematics of the split-optic propagation is given in the inset illustration. (a) shows the propagation through the glass of the first 2 cm thick optic (1^st glass) at z = 0 and z = 2 cm, and is common to the different air-gap size cases simulated, with varying L_gap value: (b) 0, (c) 2 cm, (d) 3 cm, (e) 6 cm, (f) 9 cm (I₀ = 8 GW/cm², a = 0.25, W = 0.2 mm, L_in = 2 cm, I_fil = 150 GW/cm² = 18.75·I₀). The z value where the snapshots are given are indicated in a gray box above each intensity distribution.

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As the airgap sizes exceeds L_gap=2cm, the behavior deviates from the simplified single hot-spot behavior, as interference of perturbations in the airgap becomes more significant. When the airgap L_gap=3cm (Fig. 2(d)) the interference results at the entry to the 2^nd glass (at z = 5cm) in secondary side lobes intertwining the main ones, that originally been launched into the airgap. The increase in the number of peaks at the entrance of the nonlinear media (2^nd glass) leads to a reduction in the peak intensities, for power conservation considerations, and this reduction in the perturbation amplitude results in an increased distance till collapse [19], L_fil=8.3cm. However, for further increase in the airgap the interference pattern evolves such that the secondary side-lobes intensify while the originally main lobes weaken and disappear, and for the case of L_gap=6 cm, illustrated in Fig. 2(e), a significant portion of the power is comprising what were the secondary side-lobes. As the airgap size increases from 3cm to 6cm, and the power is transitioning more into the second side-lobes, L_fil values constantly decrease reaching L_fil=3.47cm at L_gap=6cm, which is about the same value as without an airgap.

The L_fil minima observed in Fig. 1(c) at about L_gap=5.75cm could be referred to reaching half the Talbot distance in the airgap, where the first Talbot imaging plane is predicted, and indeed also shifted in space half a period (using the diagonal period = $W/\sqrt 2$, the half Talbot distance is ${W^2}/(2{\lambda _0})$=5.7 cm, in agreement with the minima data) [23]. Following the Talbot effect, increasing further the gap size, the interference evolves back to the airgap injected hot-spot, where in the midterm the power re-distributes back from the secondary lobes to the original lobes. At some airgap value in between, illustrated for L_gap=9cm in Fig. 2(f), both main and substantial side lobes co-exist, resulting in reduced peak intensity, and thus enhanced L_fil=7.44.

Clearly, the nature of this Talbot based interference process, outlined in this section for periodic perturbation, leads to a periodic and not a monotonically growth with the airgap size. Nevertheless, the more prototypical perturbation due to intensity contrast on large beam has random and more than one spatial frequency spectral content. Yet, the limited L_fil that could be obtained by this airgap mitigation due to interference and thus limited reduction in the hotspots entering the 2^nd glass is expected to play a role in the more realistic case, as discussed in the following section.

3. Random harmonic perturbation

To study the more realistic case of intensity contrast on a flat-top beam, we study a randomly synthesized excitation source with a given periods spatial-spectral content. A 2D white noise was generated with uniform random distribution of spatial spectrum (on a base beam with given intensity I₀), followingly it was band-pass filtered (BPF, with central period W, and a period content band of ΔW around it), and then the amplitude root mean square (RMS) was normalized to the specified amplitude value, a (with respect to the base beam amplitude). The calculation grid size used is 2.6 mm on 2.6 mm for the calculations presented in this section. The effect of changing the grid size and of the variance in the results due to the stochastic nature of the excitation was studied and was found to be significantly smaller than the main effects to be discussed hereafter in this section (see details in Appendix A).

An example for the beam evolution through the airgap split glass configuration is illustrated in Fig. 3, for a 7 cm airgap. The self-focusing is observed in both glass sections as the peak intensity increases and hotspots are narrowing with further propagation in the medium. In the airgap section the peak intensity is increased at the first 0.5 cm and reaches 2.6·I₀ (with respect to the 2.2·I₀ at airgap entrance), then for about additional 2.5 cm the peak intensity rapidly decreases to 2.2·I₀ and the hot-spots expands, followed by a slowed down peak intensity decrease for the remaining 4 cm of airgap (see also Visualization 2). These consecutive three regions of behavior could be attributed to the 1^st glass lensing; the ‘freely’ diffracting hotspots; and the interference propagation regions; accordingly.

 

Fig. 3. Normalized intensity distribution (I/I₀) propagation snap-shots as a function of distance through the split-optic system for a flat-top beam with a random perturbation noise source (L_gap = 7 cm, I₀ = 8 GW/cm², a = 0.25, W = 0.2 mm, ΔW = 0.1 mm, L_in = 2 cm). (see Visualization 2). This figure shows the propagation through the glass of the first 2 cm thick optic (1^st glass) at z = 0 and z = 2 cm, propagation through the airgap from z = 2 cm to z = 9 cm, and propagation through the glass of the second 2 cm thick optic (2^nd glass) at points z = 11 cm and z = 11.5 cm. For each propagation snap-shot, the z-value and the maximal value of I/I₀ range, I_max, is given at the gray box above the intensity distribution.

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Studying the airgap split glass with random BPF contrast excitation, shows a substantial (about >30%) but limited enhancement in L_fil with increasing the airgap. In the test case illustrated in Fig. 4(a), L_fil is plotted as a function of airgap, for central BPF period of W = 0.2 mm, and for three different period content bands, ΔW. Common to all these three cases, is an increase in L_fil with the airgap size for the relatively short ones, while further increase seems to result in about the same L_fil value of around 4.75 cm. While the saturated increase value in L_fil value is about the same for these 3 cases, the airgap value at which it is being reached (L_gap^sat) varies, reducing as ΔW increases. In Fig. 4(b), the results for a similar analysis but with a different mean period W = 0.3 mm is presented. The same behavior is observed, as the saturated improvement in L_fil value is reached, however L_gap^sat obtains larger values. The observed increase in L_gap^sat with increase in W (between the two cases shown in Figs. 4(a) and 4(b), and with the reduction in ΔW (seen per each one of these two cases) suggests that L_gap^sat is dominated mainly by the shorter periods of the excitation source.

 

Fig. 4. L_fil as a function of airgap for random BPF contrast excitation, with average period: (a) W = 0.2 cm, and (b) W = 0.3 cm, for different period content band, ΔW, as specified at the legend. (I₀ = 8 GW/cm², a = 0.25, L_in = 2 cm).

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A broader parametric study of L_fil as a function of W and L_gap, for a random BPF contrast excitation corroborates for a larger set of cases the saturation of L_fil above some L_gap^sat air gap size, which increases with W (Fig. 5(a)). The self-collapse distance at the 2^nd glass (and thus also the total one in glass, L_fil) depends mainly on the base beam intensity, and the seed perturbation amplitude and period [19]. Therefore, for given I₀ and content of spatial periods (constant W lineout in Fig. 5(a)), the constant L_fil above some L_gap^sat suggests also constant mean perturbation amplitude (i.e., intensity contrast) at the 2^nd glass entrance. The “thermalization” of the beam intensity modulation statistics to a constant contrast after some transitional propagation in the airgap could be associated with the nature of Fourier optics [24,25]. Since the airgap is a linear medium, the spatial-spectral content of the beam is unchanged within it, and therefore it is determined by the excitation and by the propagation in the nonlinear medium of the 1^st glass. Even though the amplitude of the spatial spectrum is constant in the airgap, the accumulated phase rate of each component is different, and therefore the interference pattern entering the second slab varies. After enough accumulated phase cycles have passed during propagation, the multiple contributions add incoherently, leading to a “thermalized” statistic with the amplitude contrast converging to the square root of the power spectral density (PSD), a result of the Fourier transform properties. The increase in L_gap^sat with W, observed in Fig. 5(a), is a result of the slower diffraction for larger period features (small components, generate “faster” waves, propagating at larger angles off-axis, thus at a given axial propagation distance they acquire more accumulated phase cycles than a “slower” wave).

 

Fig. 5. (a) L_fil as a function of airgap and of W for random BPF contrast excitation; and (b) comparison of the mean and maximal value of L_fil over all the airgap runs for each W case to the BT-type predicted L_fil using periodic excitation at W without airgap (I₀ = 8 GW/cm², a = 0.25, L_in = 2 cm, ΔW = 0.1 mm).

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The change in W also leads to a change in maximal L_fil obtained. This is clearly observed in Fig. 5(a), as highest L_fil values in the figure (yellow contours) are reached for the higher W values. To better examine the dependence of the values at saturation, for each W value we plot the mean and the max over the entire examined airgap range (see Fig. 5(b), in blue symbols). Varying the mean period leads to a different perturbation growth at the 2^nd glass, following Bespalov-Talanov gain (BT-gain) [26–28]. The distance to self-focusing was shown elsewhere [19] to follow small-signal BT-gain like behavior, even though the collapse process deviates substantially from the linearized BT assertions. The BT-gain-like behavior is represented by the L_fil measured without airgap and with periodic excitation at the given W (red circles), showing the same trending behavior as that of the saturation value.

Additional studies of the dependence on the period spectral bandwidth of the excitation source, further corroborates that the shortest periods predominantly determine the behavior. In Fig. 6(a), L_fil is shown as a function of the period bandwidth and airgap size, for a constant mean period, W = 0.2 mm (and in Fig. 6(b), cross sections of Fig. 6(a) are given for several ΔW values, for clearer comparison of L_fil values). It is observed that L_gap^sat reduces as ΔW increases, which is consistent with the results derived based on Fig. 5(a), since as ΔW value increases the shortest period in the period-spectrum reduces (similar effect to the reduction of W, as shown in Fig. 5). Furthermore, in Figs. 6(c) and 6(d), we find that band-pass filtering and low-pass filtering (LPF) behaves similarly, when comparing L_fil as a function of the airgap size and the smallest period included, W_c (i.e., cut-off period). This further support the short period being predominant in determining L_gap^sat.

 

Fig. 6. (a) L_fil as a function of airgap and of ΔW for random BPF contrast excitation, and (b) its cross-sections for several given ΔW values. Comparison of BPF and LPF, for L_fil as function of its cutoff period, W_c, and airgap size. (I₀ = 8 GW/cm², a = 0.25, L_in = 2 cm, for BPF case: W = 0.2 mm).

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The L_fil reduces as the intensity contrast increases and as the base-beam intensity increases. The L_fil is depicted as a function of a and L_gap in Fig. 7(a), and in Fig. 7(b) the mean value of L_fil over the L_gap studied range for each a is plotted (I₀ = 8 GW/cm²). A similar study for I₀ is depicted accordingly in Figs. 7(c) and 7(d) (a = 0.25). These four plots elucidate that there is a similar monotonic dependence of the L_fil on both I₀ and a. This dependence is consistent with the known growth of filamentation in self-focusing media [19], and as a result of the airgap not changing the base-beam intensity and the intensity contrast for airgaps beyond L_gap^sat.

 

Fig. 7. Dependence of L_fil on excitation perturbation amplitude and base beam intensity: L_fil as a function of airgap and (a) a, and (c) I₀; and averaged L_fil over airgap range as a function of (b) a, and (d) I₀; (e) I₀ as a function of L_fil in which its achieved for two cases: without airgap (blue), and 5 cm airgap (red). (I₀ = 8 GW/cm², a = 0.25, L_in = 2 cm, for BPF case: W = 0.2 mm).

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The permissible base beam intensity before reaching collapse for a given glass path-length could be substantially increased by applying an airgap split. As a study-case the comparison in permissible base beam intensity before collapse was analyzed for 5 cm airgap with comparison to the no airgap reference case. A given beam contrast, a = 0.25, and a random BPF period bandwidth was assumed, W = 0.2 mm, ΔW = 0.1 mm. Using the data in Fig. 7(a), for the two L_gap values of 0 and 5 cm, I₀ are plotted as a function of the resulting L_fil, in Fig. 7(e). When choosing, for example, a given glass length of 4 cm (the length could be attributed to an optical element we wish to avoid damaging), the permissible beam intensities increase by about 30%, which is significant. Since the chosen 5 cm airgap is at the saturation values region, it is also representative of what would be achieved by choosing larger airgap values. Furthermore, this substantial increase is also present for other glass lengths, as could be viewed in this figure (see other L_fil values).

Reducing the first glass length enhances a local peak behavior in L_fil, as evident in Fig. 8, in between the ‘rise’ and ‘saturation’ regimes. A possible interpration is that the pre-airgap entry self-focusing propagation in the first glass supresses the side lobes, as the main lobes are enhanced substantially more due to the non-linear nature of the process. These side-lobes may evolve into additional peaks while the interference at the airgap is still not randomized, and result in a local peak behavior, as was discussed also for the periodic case in Section 2. For the cases of the smaller L_in, this local peak behavior in L_fil terminates at the entry to the saturation distance into the airgap (about 3-4 cm, for this case, determined by the period-content, as discussed before) with lower L_fil values. This might point to an advantage is scheme where the initial glass is not too thin. This would lead to the same practical implementation resulting of the more intuitive wisdom based on the emphirical observation that the multiplication of the beam base intensity and the distance to collapse is a constant (otherwise known as the IL-rule) [29,30], thus for maximizing the permisible intensity in the two segments problem, the glass path-length should be split equally between two segments.

 

Fig. 8. (a) L_fil as a function of airgap and 1^st glass length, and (b) its cross-sections for several given L_in values. (I₀ = 8 GW/cm², a = 0.25, L_in = 2 cm, for BPF case: W = 0.2 mm).

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4. Conclusion

Mitigation of filamentation by splitting a thicker optic into two thinner optics spaced by airgap is numerically studied in the framework relevant to high-power laser system and shown to be effective yet limited by the interference dynamics of the beam intensity contrast noise while propagating in the airgap. Unlike the whole beam / isolated perturbation case that is being more commonly considered, where improvement monotonically increases with airgap size, for the more realistic intensity contrast on a flat-top case, the interference in the airgap limits the improvement of the effect, in terms of propagation distance in glass till collapse (Lfil).

As much as a 30% percent improvement in the glass length before filamentation is demonstrated by introducing an airgap. This may have potential implication on high-power laser final optics design (e.g., few cm thick optical elements of Mega-Joule laser systems). The case study example suggests that splitting a 4 cm optics into two thinner elements, spaced by a 5 cm airgap enhances the permissible intensities and thus also the power by about 30%, which is substantial. Furthermore, the mechanisms presented here show that as long as the airgap is chosen in the saturation range it is relatively non-sensitive to the exact air-gap selected value, and serves as an effective mitigation to a large span of contrast spatial periods. We note that the optics considered here consist of flat slabs, where in practice the optics may have curved surfaces, which would modify the phase front of the interfering hot spots. As long as the introduced phase-front modification is slow enough it would enter only secondary corrections, yet this effect may deserve a future exploration. While many high-power laser systems use a flat top profile (which for a given system clear aperture, maximizes the beam power for laser-induced damage limited peak power, and also maximizes the gain slab energy extraction efficiency), other high-power laser systems use other profiles for other optical design considerations (such as optimizing intensity on target by using diffraction limited profiles). At the scale of the perturbations, the beam profile is typically slowly changing, and therefore the analysis here of perturbation on a flat base-beam should serve as a good approximated model, where the beam’s curvature could be thought of as a very large period perturbation, thus as showed here not dominating the problem.

The maximum improvement in Lfil depends on the base beam intensity, the magnitude of the beam contrast, and the smallest spatial frequency in the noise. On the other hand, the larger the perturbation period in the noise is, the smaller its growth is [19]. Therefore, a practical approach is to determine the largest period that might lead to filamentation in the considered optic (since the BT-gain declines with increasing the period), and use the saturation threshold airgap length (Lgapsat) for it (since shorter periods should already be saturated at this Lgapsat).

Appendix A: dependence of results on the grid size

In Fig. 9(a), L_fil is plotted as a function of airgap, for a given calculation scenario (I₀ = 8 GW/cm², a = 0.25, L_in = 2 cm, W = 0.2 mm, ΔW = 0.1 mm), for different grid sizes and repeats of randomly generated excitation. The difference resulting from the randomness in the source altogether with the variation in grid size is significantly smaller than the main effect discussed in the paper, and also clearly observed in the plot.

 

Fig. 9. (a) L_fil as a function of airgap, for different calculation grid-size and initial excitation. For each calculation scenario (I₀ = 8 GW/cm², a = 0.25, L_in = 2 cm, W = 0.2 mm, ΔW = 0.1 mm) and given air-gap size, 24 calculation are conducted and overlaid: 8 different grid size (linear spacing between 1.5 and 2.5 mm) × 3 initial conditions (to examine the effect of the excitation randomness) represented by the 3 colors. (b)) L_fil as a function of calculation grid-size, for one airgap case (L_gap=0). 3 repeats of the calculation are given at different colors.

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This numerical artifact is a result of the finiteness of practical calculation grid-sizes, thus is hard to avoid. An alternative approach is absorbing boundary conditions (ABL) – however, testing this alternative approach we found that designing the roll-off slope of the ABL to not dominate the filamentation at the edge or corner, or to create substantial beam-edge ringing, is even more challenging.

The three colors in the plot identify 3 repeats of the calculations, to illustrate how the stochastic seed affect the results, and the different grid-size are dots of the same color. Even with the spread around the data curve (combined result of randomness and grid-size variation), the prototypical behavior discussed in the paper is clearly observed: an increase in L_fil for the smaller airgaps and reaching a saturation value for further increasing the airgaps. While the entire shift due to the physical effect is about 1.5 cm (being an observation in this paper), the STD due to the numerical artifacts is much smaller (about 0.1 – 0.3 cm).

The graph in Fig. 9(b) details the dependence of the filamentation length as a function of the grid size for the 3 repeats, for one airgap size case (no airgap). It is clear that the grid-size does not have a systematic parametric dependence, but rather has an effect of enhancing the randomness of the result. However, the observations and interpretations presented in this study focus on systematic observations (identifiable trends when changing parameters, being consistent across multiple seed cases and various parameter settings), which are also consistent with identified underlying physic mechanisms.

Acknowledgments

This work was performed under the auspices of the U.S. Department of Energy by Lawrence Livermore National Laboratory under Contract DE-AC52-07NA27344. LLNL-JRNL-787919.

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